I can not resolve an issue of the book Mathematical Statistics of Shao, is as follows:
If $E|X_{1}|^3$ is finite, get $E(\bar{X}^3)$ and $Cov(\bar{X},S^2)$
If $E|X_{1}|^4$ is finite, get $Var(S^{2})$
My question is how to write $\bar{X}^3$ to calculate the expected value? The other I know to do. I thank you!
I tried to write $\displaystyle{(\sum(X_{i}))^3=\sum(X_{i}^{3})+3*\sum_{i=1}^{n}\sum_{i\neq j=1}^{n}X_{i}X_{j}^{2}}$ but I saw what is wrong. I also tried to do $\displaystyle{(\sum(X_{i}))^3=\sum_{i}\sum_{j}\sum_{k}X_{i}*X_{j}*X_{k}}$ and $\displaystyle{(\sum(X_{i}))^3=\sum_{i}X_{i}*(\sum_{i}X_{i}^2+\sum_{1\leq i < j \leq n}X_{i}X_{j})}$ but I saw what is wrong.